Emergent Newtonian Gravity from FCC Lattice Phonon Dynamics via Inclusion Mechanics

This paper derives Newton's gravitational constant G as an emergent phenomenon arising from the elastic mechanics of a face-centred cubic (FCC) Planck-scale lattice substrate — the Field of Resonance (FoR) framework. Version 6 supersedes v5 with two substantive additions to the Discussion and Open Problems sections: Addition 1 — Analytic determination of k₀ (Open Problem 1 advancement). The ground-state wavevector k₀ is shown analytically to be determined by the substrate field parameters via k₀ = k_peak × (U₀α_k/σ²) / (2c₀² + U₀α_k/σ²), verified across 1,980 independent parameter combinations to 0.2% accuracy. k₀ is not a free parameter — it is derived from the field theory. k_peak, the inverse interaction range of the substrate field, is identified as the one remaining free parameter at Level 1, with a clear physical identity. Addition 2 — Holographic partial resolution of Open Problem 2. The Bekenstein-Hawking bound applied to a Planck-scale cubic cell constrains the substrate field interaction range to k_peak = (9π²)^{1/3}/ℓ_Pl = 4.462/ℓ_Pl, in agreement with the FoR geometric requirement k_peak = √2π/ℓ_Pl = 4.443/ℓ_Pl to 0.43%. The FoR value sits at the centre of the holographically allowed window. No fine-tuning is required: all dimensionless field parameter ratios are of order unity at this scale. The 0.43% residual is attributed to the cube-versus-truncated-octahedron geometry approximation of the FCC Wigner-Seitz cell. All core results from v5 are unchanged: G = c³ℓ₀²/ħ confirmed by four independent routes, α = 9/2 from the FCC elastic tensor via collective wave-packet anharmonicity, the Cauchy relation C₁₂ = C₄₄ exact for central-force FCC, N-cancellation proven, equivalence principle derived. The complete substrate Lagrangian derivation is published separately as Paper 3 of the FoR series.